Machine design and electrical contacts involve frequently elastic circular contacts subjected to normal loads. Depending on geometry, these may be Hertzian or surface contacts. Both possess highly nonuniform pressure distributions which diminish contact load carrying capacity. The achievement of a uniform pressure distribution would be ideal to improve the situation, but this violates stress continuity. Instead, the generation of a uniform pressure over most of contact area can be sought. Generally, equivalent punch profile which generates this pressure is found by numerical evaluation of double integrals. This paper simplifies the derivation of punch profile by using an existing correspondence between a polynomial punch surface and elastically generated pressure. First, an improved pressure profile is proposed seeking to avoid high Huber-Mises-Hencky stresses near contact surface. Then, this is approximated by the product between typical Hertz square root and an even polynomial, which yields directly the punch profile. Formulas for normal approach and central pressure are derived.

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